it is universal in the usual sense of the word. but embedding a (small) lie algebra into an (infinite dimensional) quotient of a tensor algebra does not endow the original lie algebra with a 'naturally occuring' product, does it? the product so given is not (necessarily) defined on the lie algebra before you make this construction. the example i gave of C^3 with relations forcing it to be sl_2 shows that. C^3 comes with no 'natural' algebra structure for which [ef]=ef-fe.Don Aman said:It seems you consider the universal enveloping algebra "contrived", but as it is a universal functor from the category of Lie algebras, it is natural in the usual sense of the word.
we can embed (finitely generated) abelian monoids into (finitely generated abelian) groups (at least thati s my gut reaction, i can't say i've proved it, but it seems clear), that doesn't mean that there isn't a naturally occuring monoid that isn't a group, does it?
and the question was if there was a naturally occuring situation when we *do* turn a blind eye to it, or more accurately when we plain don't see it to begin with (like N in Z). why is that so hard to grasp? i can't think of one, and you can't either, apparently, so why do you keep telling me you can't think of one?But every Lie algebra has a multiplication, and the only way you'll find one without multiplication is if you turn a blind eye to it.
so now you can think of one? which is it?That's why I mention the Lie algebras of Lie groups that aren't matrix groups (say, Spin(n) for a big enough n). The only way it'll have a multiplication is if you go out of your way to define one.
As another example I know fundamental group of a space can be made arbitrary, that doesn't mean every group is a fundamental group, it is merely isomorphic to one. Is the distinction clear?